The Constant in the Theorem of Binev-Dahmen-DeVore-Stevenson and a Generalisation of it

TL;DR

This paper improves bounds on the ratio of marked to bisected simplices in recursive triangulation refinements, generalizes initial conditions, and extends existing theorems to higher dimensions with practical initializations.

Contribution

It introduces a new bound for triangulation refinement ratios, generalizes initial conditions, and extends the Binev-Dahmen-DeVore theorem to any dimension with practical initializations.

Findings

Bound on the ratio of marked to bisected simplices is improved.

The result applies under weaker initial conditions.

Number of recursive bisections per simplex is bounded by twice the dimension.

Abstract

A triangulation of a polytope into simplices is refined recursively. In every refinement round, some simplices which have been marked by an external algorithm are bisected and some others around also must be bisected to retain regularity of the triangulation. The ratio of the total number of marked simplices and the total number of bisected simplices is bounded from above. Binev, Dahmen and DeVore proved under a certain initial condition a bound that depends only on the initial triangulation. This thesis proposes a new way to obtain a better bound in any dimension. Furthermore, the result is proven for a weaker initial condition, invented by Alk\"amper, Gaspoz and Kl\"ofkorn, who also found an algorithm to realise this condition for any regular initial triangulation. Supposably, it is the first proof for a Binev-Dahmen-DeVore theorem in any dimension with always practically realiseable initial conditions without an initial refinement. Additionally, the initialisation refinement proposed by Kossaczk\'y and Stevenson is generalised, and the number of recursive bisections of one single simplex in one refinement round is bounded from above by twice the dimension, sharpening a result of Gallistl, Schedensack and Stevenson.